Methods and apparatus for positioning aircraft based on images of mobile targets

ABSTRACT

Methods and apparatus for positioning aircraft based on images of mobile targets are disclosed. An example method includes identifying, by executing first instructions via a processor, a mobile target in an image obtained by a camera mounted on a first aircraft and obtaining, by executing second instructions via the processor, current coordinates of the first aircraft. The example method includes determining, by executing third instructions via the processor, coordinates of the mobile target based on the coordinates of the first aircraft and the image. The coordinates of the mobile target are within an area of uncertainty. The example method includes determining, by executing fourth instructions via the processor, a first position for the first aircraft that reduces the area of uncertainty of the coordinates of the mobile target.

CROSS-REFERENCE TO RELATED APPLICATIONS

This patent claims priority to European Application Number EP15382382,filed on Jul. 23, 2015, which is incorporated herein by reference in itsentirety.

FIELD OF THE DISCLOSURE

This patent relates generally to positioning aircraft and, moreparticularly, to methods and apparatus for positioning aircraft based onimages of mobile targets.

BACKGROUND

Mobile objects (e.g., people, land vehicles, water vehicles, etc.) areoften tracked by aircraft. Recently, unmanned aircraft have beenutilized to track the mobile objects. These unmanned aircraft includecameras that enable the mobile objects to be identified and/or tracked.In some examples, the unmanned aircraft are planes that circle thetracked mobile objects and/or perform other maneuvers to track themobile objects.

SUMMARY

In one example, a method includes identifying, by executing firstinstructions via a processor, a mobile target in an image obtained by acamera mounted on a first aircraft and obtaining, by executing secondinstructions via the processor, current coordinates of the firstaircraft. The method includes determining, by executing thirdinstructions via the processor, coordinates of the mobile target basedon the coordinates of the first aircraft and the image. The coordinatesof the mobile target are within an area of uncertainty. The methodincludes determining, by executing fourth instructions via theprocessor, a first position for the first aircraft that reduces the areaof uncertainty of the coordinates of the mobile target.

In another example, an apparatus includes a camera mounted to anaircraft to obtain an image. The apparatus includes a processor of theaircraft to identify a mobile target in the image obtained by thecamera, obtain current coordinates of the aircraft, and determinecoordinates of the mobile target based on the coordinates of theaircraft and the image. The coordinates of the mobile target is withinan area of uncertainty. The processor of the aircraft is to determine aposition for the aircraft that reduces the area of uncertainty of thecoordinates of the mobile target and instruct the aircraft to move tothe position to reduce the area of uncertainty.

In another example, an apparatus includes means for obtaining an imagemounted to an aircraft. The apparatus includes means for determining aposition of the aircraft to identify a mobile target in an imageobtained by the means for obtaining an image, obtain current coordinatesof the aircraft, determine coordinates of the mobile target and an areaof uncertainty based on the coordinates of the aircraft and the image,and determine a position for the aircraft that reduces the area ofuncertainty. The apparatus includes means for moving the aircraft to theposition to reduce the area of uncertainty.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example aircraft in accordance with the teachingsof this disclosure.

FIG. 2 illustrates an example camera coupled to the aircraft of FIG. 1.

FIG. 3 illustrates the aircraft of FIG. 1 positioned to reduce anexample area of uncertainty for coordinates of an example mobile targetin accordance with the teachings of his disclosure.

FIG. 4 illustrates the aircraft of FIG. 1 and another aircraftpositioned to further reduce the area of uncertainty for the coordinatesof the mobile target of FIG. 3 in accordance with the teachings of hisdisclosure.

FIG. 5 depicts a global positioning system utilized to determinecoordinates of the aircraft of FIGS. 1-4, the other aircraft of FIG. 4and/or the mobile target of FIGS. 3-4.

FIG. 6 is a flowchart representative of an example method to positionthe aircraft of FIGS. 1-4 and/or the other aircraft of FIG. 4 to reducethe area of uncertainty for the coordinates of the mobile target ofFIGS. 3-4 in accordance with the teachings herein.

The figures are not to scale. Instead, to clarify multiple layers andregions, the thicknesses of the layers may be enlarged in the drawings.Wherever possible, the same reference numbers will be used throughoutthe drawing(s) and accompanying written description to refer to the sameor like parts.

DETAILED DESCRIPTION

Mobile objects are often tracked by aircraft. For example, mobileobjects that travel along land (e.g., people, animals, land vehicles,etc.) and/or water (e.g., people, animals, water vehicles, etc.) may betracked by aircraft. Unmanned aircraft are now being utilized to trackmobile objects. In some examples, the unmanned aircraft are pilotedand/or controlled by a person in a remote location (e.g., via radiocontrol and/or a human-machine interface). Other example unmannedaircraft automatically track and/or follow mobile objects without humanintervention (e.g., without a pilot within and/or remote from theaircraft). To enable such unmanned aircraft to identify and/or track themobile objects, the unmanned aircraft include cameras that obtain imagesof the mobile objects. Some known unmanned aircraft that track mobileobjects without human intervention include planes with cameras thatobtain images of the mobile objects. In some instances, the unmannedplanes may perform complicated maneuvers to continue to track the mobileobjects. For example, because planes must move continuously, theunmanned planes may attempt to circle around (e.g., in an ellipticaltrajectory) a mobile object that becomes stationary and/or moves at aslow rate (e.g., slower than the unmanned planes is able to travel). Insome instances, the maneuvers performed by the unmanned plane cause theunmanned plane to lose and/or determine inaccurate coordinates of themobile object.

The example apparatus and methods disclosed herein determine a position(e.g., a stationary position) and/or orientation at which an unmannedaircraft is able to accurately determine and/or track coordinates of amobile target. For example, the apparatus and methods disclosed hereindetermine a position and an orientation at which an unmanned helicopterand/or other unmanned rotocraft may be able to be positioned (e.g., byhovering) to accurately track a mobile target without having to performdifficult flight maneuvers. Further, by monitoring the mobile targetfrom the stationary position, the example methods and apparatusdisclosed herein reduce an amount of processing performed (e.g., by aprocessor such as a microcontroller or microcontroller unit) todetermine coordinates of the mobile target by collecting data (e.g.,from images obtained from a camera of the aircraft) from a stationary(e.g., not moving) reference point.

The examples disclosed herein include identifying a mobile target in animage obtained by a camera (e.g., a first camera) mounted on an aircraft(e.g., a first aircraft) and obtaining current coordinates of theaircraft. For example, the current coordinates are obtained from datareceived via a global positioning sensor in communication with theprocessor of the aircraft. Further, in the examples disclosed herein,coordinates are determined for the mobile target based on thecoordinates of the aircraft and the image obtained from the camera. Thecoordinates of the mobile target are within an area of uncertainty(e.g., resulting from margins of error of the obtained data). Further, afirst position and/or first orientation for the aircraft are determinedthat reduce (e g, minimizes) the area uncertainty of the coordinates ofthe mobile target when the aircraft is tracking the mobile target. Insome examples, the aircraft is instructed to move to the first positionand/or first orientation (e.g., via an autopilot system of theaircraft). For example, the aircraft is moved to the first position toenable the camera of the aircraft to obtain a second image in which themobile target is centered (e.g., is in a principal position) to furtherreduce (e.g., minimize) the area of uncertainty in tracking the mobiletarget.

In some examples, the coordinates of the mobile target are 3-dimensionalcoordinates that are determined based 3-dimensional referencecoordinates of the aircraft and the 2-dimensional image obtained by thecamera of the aircraft. Additionally or alternatively, a distancebetween the aircraft and/or the camera of the aircraft is determined andutilized to determine the coordinates of the mobile target. Further, asize, a color and/or a velocity of the mobile target may be determinedbased on the image obtained by the camera to determine a classificationof the mobile target to further reduce the area of uncertainty and/orenable the aircraft to track the mobile target. In such examples, avector is formed that includes information related to the mobile targetsuch as size, color, velocity, weight, class, position, etc.

The first position of the aircraft may be based on the coordinates ofthe aircraft, the coordinates of the mobile target, the image obtainedby the camera, a focal length of the camera, an angle of the camerarelative to a fuselage of the aircraft and/or a yaw of the aircraft.Further, the area of uncertainty of the coordinates of the mobile targetmay have an elliptical shape that includes a major axis and a minoraxis. For example, the first position of the aircraft that reduces thearea of uncertainty is orthogonal to the major axis and/or along theminor axis. In some examples, two positions are identified for theaircraft that would minimize the area of uncertainty and are orthogonalto the major axis and/or along the minor axis. In such examples, theaircraft is instructed to move to the position that is closest to thecurrent coordinates of the aircraft.

In some examples, the mobile target is identified and/or tracked basedon a second image obtained by another camera (e.g., a second camera)mounted on another aircraft (e.g., a second aircraft). In such examples,current coordinates of the other aircraft are obtained and a secondposition for the other aircraft is determined that would further reducethe area of uncertainty of the coordinates of the mobile target. Forexample, the second position for the other aircraft is orthogonal to theminor axis and/or along the major axis of the area of uncertainty whenthe first position of the initial aircraft is orthogonal to the majoraxis and/or along the minor axis.

Further, in some examples, images of a mobile target are analyzed toposition aircraft by identifying a mobile target in an image obtainedvia a camera mounted on an aircraft, determining (e.g., via internalsensors in communication with the processor of the aircraft) coordinatesof the aircraft with respect to a Global Reference System, determiningcoordinates of the mobile target with respect to the Global ReferenceSystem, determining an area of uncertainty associated with thecoordinates of the mobile target, and determining a first position ofthe aircraft from which to monitor the mobile target that minimizes thearea of uncertainty associated with the coordinates of the mobiletarget. For example, to determine the coordinates of the mobile targetwith respect to the Global Reference System, a distance between theaircraft and the mobile target is determined (e.g., via a sensor such asa telemeter and/or a laser telemeter that is mounted on the aircraft andin communication with the processor), coordinates of the mobile targetare determined with respect to axes of the image of the camera, an angleof the camera relative to a fuselage of the aircraft (e.g., a fixedangle, an adjustable angle) is determined, an orientation of theaircraft (e.g., a yaw angle) relative to the Global Positioning Systemis determined (e.g., via a sensor, such as a gyroscope, of the aircraftthat is in communication with the processor of the aircraft), and thecoordinates of the mobile target in the image (e.g., 2-dimensionalcoordinates) are transformed into coordinates of the mobile targetrelative to the Global Positioning System (e.g., 3-dimensionalcoordinates).

Further, in some examples, a real-time position of an aircraft of afleet of aircraft may be known, in real-time, to the other aircraft ofthe fleet to enable the aircraft to be positioned relative to each otherin an efficient manner for tracking and/or monitoring the mobile target.For example, the aircraft of the fleet that is closer and/or is able tomore easily move to the first position to monitor the mobile devicemoves to the first position. Further, the fleet of aircraft may takeinto account characteristics of the aircraft (e.g., maximum velocity,fuel level, etc.) when determining which aircraft is to move to thefirst position. If it is determined that an aircraft is not in acondition to track a mobile target, the aircraft may be instructed toreturn to a base to address and/or resolve the conditions of theaircraft.

Thus, the examples disclosed herein enable aerial tracking of a mobiletarget by an unmanned aircraft that remains suspended at a fixedposition without operator intervention by determining a position atwhich the aircraft is able to monitor the mobile target with a leastamount of uncertainty with respect to determined coordinates of themobile target.

FIG. 1 illustrates an example aircraft 1 in accordance with theteachings herein. In the illustrated example, the aircraft 1 is ahelicopter and/or another type of rotocraft. As illustrated in FIG. 1, afuselage 4 of the aircraft 1 is oriented in relation to an â axis, a{circumflex over (b)} axis, and a ĉ axis. The orientation of thefuselage 4 about the â axis defines a roll angle of the aircraft 1, theorientation about the {circumflex over (b)} axis defines a pitch angleof the aircraft 1, and the orientation about the ĉ axis defines a yawangle of the aircraft 1. In the illustrated example, the roll angle, thepitch angle, and the yaw angle are approximately 0 degrees.

FIG. 2 illustrates a camera 3 of the aircraft 1 that is mounted,attached and/or otherwise coupled to the fuselage 4 of the aircraft 1.The camera 3 obtains images of land/or water below the aircraft 1 toenable the aircraft 1 to identify and/or track mobile objects (e.g., amobile target 2 of FIGS. 3 and 4) travelling along the land and/orwater. In the illustrated example, the camera 3 is coupled to a lowerportion of the fuselage 4 to enable the camera 3 to obtain imageswithout obstruction from the fuselage 4. For example, the camera 3 has afocal length that affects the images obtained by the camera 3. Further,the camera 3 is oriented at an angle θ relative to the fuselage 4 of theaircraft 1 to affect what is included in the images obtained by thecamera 3. In the illustrated example, the camera 3 is orientated at theangle θ such that an image obtained by the camera 3 extends along a ûaxis and a {circumflex over (v)} axis. Further, in some examples inwhich the camera 3 is oriented relative to the ground at anon-perpendicular angle, the image obtained by the camera 3 may beskewed.

FIG. 3 illustrates the aircraft 1 at a first position to track anexample mobile target 2 in accordance with the teachings of hisdisclosure. More specifically, the aircraft 1 is positioned to reduce(e.g., minimize) an area of uncertainty 5 associated with calculatedcoordinates of the mobile target 2 to enable the aircraft 1 toaccurately track and/or monitor the mobile target 2 from a stationaryposition.

For example, to reduce the area of uncertainty 5 associated with thecoordinates of the mobile target 2, the mobile target 2 is identified inan image obtained by the camera 3 and current coordinates of theaircraft 1 are obtained from a sensor (e.g., a global positioningsensor) of the aircraft 1. Coordinates of the mobile target 2 arecalculated and/or determined based on the current coordinates of theaircraft 1 and the image obtained by the camera 3. Further, the area ofuncertainty 5 associated with the determined coordinates of the mobiletarget 2 and a target position (e.g., a first position) and/ororientation (e.g., a first orientation) that reduces the area ofuncertainty 5 are determined to enable the mobile target 2 to betracked. In some examples, the aircraft 1 may be instructed to move tothe first position and/or a first orientation to reduce the area ofuncertainty 5. Additionally or alternatively, the camera 3 and sensor(s)of the aircraft 1 are in communication with a processor (e.g., amicrocontroller unit) of the aircraft to enable the processor todetermine the first position and/or the first orientation that reducesthe area of uncertainty 5 and/or to instruct the aircraft 1 to move tothe first position (e.g., via autopilot, without human intervention,etc.). For example, when the aircraft 1 is at the first position in thefirst orientation, the aircraft 1 is able to track the mobile target 2such that the mobile target 2 remains in a principal position (e.g. thecenter of the image) of the field of vision of the camera 3. In someexamples, the camera 3 is fixed relative to the fuselage 4 of theaircraft 1 without independent control of the camera 3. In otherexamples, the camera 3 camera is movable and/or controllable such thatan orientation of the camera 3 relative to the fuselage 4 is adjustable.

In some examples, there may be more than one mobile target (e.g., themobile target 2 and another mobile target) identified and/or tracked viathe images obtained by the camera 3 of the aircraft 1. In such examples,each of the mobile targets is identified by their respectivecharacteristics (e.g., shape, color, velocity, etc.). For example, thecharacteristics of the mobile targets are measured by conventionalperception systems of the camera 3 and/or the aircraft 1. In someexamples, a state vector, {right arrow over (b)}, is determined and/ordefined for each of the mobile targets and includes the identifiedcharacteristics of the respective mobile target. The state vectors ofthe respective mobile targets enable the aircraft 1 to distinguish themobile targets from each other.

For example, a state vector, {right arrow over (b_(k)(t))}, isdetermined for a mobile target, k (e.g., the mobile target 2). The statevector includes dynamic characteristic(s) (e.g., position, velocity,etc.) of the mobile target, k, and static characteristics (e.g., color,size, etc.) of the mobile target, k, that enable the mobile target, k,to be classified into a predetermined class. For example, the statevector, {right arrow over (b_(k)(t))}, is expressed as provided below inEquation 1:

$\begin{matrix}{{\overset{\rightarrow}{b_{k}}(t)} = \begin{bmatrix}{p_{k}(t)} \\{p_{k}(t)} \\\theta_{k}\end{bmatrix}} & {{Equation}\mspace{14mu} 1}\end{matrix}$

In Equation 1 provided above, p_(k) represents a position of the mobiletarget, {dot over (p)}_(k) represents a velocity of the mobile target,and θ_(k) represents static characteristics of the mobile target.

For a tracking mission in which there is a plurality of mobile targets,there is one state vector B (t)=[{right arrow over (b₁)}(t) . . . {rightarrow over (b_(k))} (t) . . . {right arrow over (b_(N))}(t)], whichincludes the respective states of each of the mobile targets 1 throughN.

The characteristics and/or information regarding the states of therespective mobile targets are determined based on measurements takenand/or data collected by a perception system of the aircraft 1 (e.g.,the camera 3 and/or other sensors) and/or the perception systems ofother aircraft (e.g., an aircraft 8 of FIG. 4) of a fleet of aircraft.The state of a mobile target is utilized to track the mobile target suchthat the mobile target is maintained in a center of an image (e.g., aprincipal position) taken by a camera of an aircraft tracking theobject. For example, the aircraft 1 utilizes the state of the mobiletarget 2 to center the mobile object 2 in an image obtained by thecamera 3 of the aircraft 1.

In some examples, margins of error in measurements (e.g., based onimages obtained by the camera 3) result in an uncertainty in theestimation of the state of the mobile target 2. In the illustratedexample, the uncertainty of a location of the mobile target 2 isrepresented by the area of uncertainty 5. The area of uncertainty 5 ofthe mobile target 2 has an associated covariance matrix, Σ. A covariancematrix is a conventional term utilized when instruments are subjected toerrors and/or tolerances in measurements and/or estimations. In theillustrated example, the covariance matrix, Σ, is an ellipsoid. Axes ofthe ellipsoid have a standard deviation, 3σ, with respect to anestimation of the position of the mobile target 2. Further, thecovariance matrix Σ is determined based on noise characteristics of thesensors of the aircraft 1 and/or methods utilized to estimate the firstposition of the aircraft 1 (e.g., noise characteristics from a positionof the camera 3 with respect the aircraft 1, a position of the aircraft1, an orientation of the aircraft 1, uncertainty of detection of themobile target 2 in image(s), etc.).

In the illustrated example, the coordinates of the first position of theaircraft 1 that minimizes the area of uncertainty 5 is represented by

${\overset{\rightarrow}{M}}_{w} = {\begin{bmatrix}x_{w} \\y_{w} \\z_{w}\end{bmatrix}.}$

The orientation of the aircraft 1 that reduces the area of uncertainty 5at the first position is represented by a roll angle, γ_(w), a pitchangle, β_(w), and a yaw angle, α_(w). Further, the current or initialcoordinates of the aircraft 1 is represented by

$\begin{bmatrix}x_{0} \\y_{0} \\z_{0}\end{bmatrix}.$

{right arrow over (M)}_(w), the roll angle, γ_(w), the pitch angle,β_(w), and the yaw angle, α_(w), are calculated based on estimatedcoordinates, {right arrow over (M)}_(t), of the mobile target 2 and thecorresponding area of uncertainty 5. The estimated coordinates, {rightarrow over (M)}_(t), of the mobile target 2 equal

$\begin{bmatrix}x_{t} \\y_{t} \\z_{t}\end{bmatrix}\quad$

in a Global Reference System (e.g., established via coordinates of theUniversal Transverse Mercator (UTM)) and are determined based oncoordinates u_(t) and v_(t) associated with pixels of the image obtainedfrom the camera 3 of the aircraft 1. For example, the coordinates u_(t)and v_(t) are expressed as a function of the axes û and {circumflex over(v)} of the image of the camera 3. Provided below in Equation 2 is abidimensional vector containing the coordinates of the image:

$\begin{matrix}{{\overset{\rightarrow}{m}}_{t} = \begin{bmatrix}u_{t} \\v_{t}\end{bmatrix}} & {{Equation}\mspace{14mu} 2}\end{matrix}$

Further, {right arrow over (m)}_(t) and {right arrow over (M)}_(t) aretransformed to {tilde over (m)} and {tilde over (M)}, respectively, asprovided below in Equation 3 and Equation 4:

$\begin{matrix}{\overset{\sim}{m} = \begin{bmatrix}u_{t} \\v_{t} \\1\end{bmatrix}} & {{Equation}\mspace{14mu} 3} \\{\overset{\sim}{M} = \begin{bmatrix}x_{t} \\y_{t} \\\begin{matrix}z_{t} \\1\end{matrix}\end{bmatrix}} & {{Equation}\mspace{14mu} 4}\end{matrix}$

The vectors {tilde over (m)} and {tilde over (M)} of Equations 3 and 4,respectively, are expressed in different reference systems and may beexpressed in relation to each other based on rotational andtransformation matrices.

For example, two reference systems {A} and {B} have a same origin ofcoordinates and standard bases {î,ĵ,{circumflex over (k)}} and{î′,ĵ′,{circumflex over (k)}′}, respectively. A vector expressed in thereference system {A} is represented as {right arrow over (v)}_(A), andthe same vector is expressed in the system {B} as {right arrow over(v)}_(B)=R_(B) ^(A){right arrow over (v)}_(A), in which R_(B) ^(A) isthe rotation matrix from the reference system {A} to the referencesystem {B}. R_(B) ^(A) is represented by Equation 5 provided below:

$\begin{matrix}{R_{B}^{A} = \begin{bmatrix}{\hat{i} \cdot {\hat{i}}^{\prime}} & {\hat{j} \cdot {\hat{i}}^{\prime}} & {\hat{k} \cdot {\hat{i}}^{\prime}} \\{\hat{i} \cdot {\hat{j}}^{\prime}} & {\hat{j} \cdot {\hat{j}}^{\prime}} & {\hat{k} \cdot {\hat{j}}^{\prime}} \\{\hat{i} \cdot {\hat{k}}^{\prime}} & {\hat{j} \cdot {\hat{k}}^{\prime}} & {\hat{k} \cdot {\hat{k}}^{\prime}}\end{bmatrix}} & {{Equation}\mspace{14mu} 5}\end{matrix}$

In examples in which free vectors are utilized, a vector may betransformed from one reference system to another reference via therotational matrix. In examples in which position vectors are utilized,the vector may be transformed via the rotational matrix and atransformation matrix. For example, if the reference systems {A} and {B}have different origin coordinates, a translation vector, {right arrowover (t)}_(A), translates an origin of {A} to an origin of {B} expressedin the system {A}, and transformation matrix, T_(A) ^(B), transforms avector expressed in the reference system {B} to a vector expressed inreference system {A} as illustrated in Equation 6 provided below:

$\begin{matrix}{\begin{bmatrix}{\overset{\rightarrow}{v}}_{A} \\1\end{bmatrix} = {{T_{A}^{B}\begin{bmatrix}{\overset{\rightarrow}{v}}_{B} \\1\end{bmatrix}} = {\begin{bmatrix}R_{A}^{B} & {\hat{t}}_{A} \\000 & 1\end{bmatrix}\begin{bmatrix}{\overset{\rightarrow}{v}}_{B} \\1\end{bmatrix}}}} & {{Equation}\mspace{14mu} 6}\end{matrix}$

Returning to the vectors {tilde over (m)} and {tilde over (M)} expressedin Equations 3 and 4, the relation between one point of Cartesian 3Dspace at which the mobile target 2 is located and its projection ontothe image obtained by the camera 3 is expressed in terms of thetransformed vectors, {tilde over (m)} and {tilde over (M)}, via Equation7 provided below:

s*{tilde over (m)}=C*T _(C) ^(G) *{tilde over (M)}=C*[R _(C) ^(G){circumflex over (t)} _(C) ]*{tilde over (M)}   Equation 7

In Equation 7, s represents a scale factor that equals a valuecorresponding to a function of the relative positions of the camera 3and the mobile target 2 and characteristics and/or parameters of thecamera 3. Further, s is defined by Equation 8 provided below:

s=(D _(t) −f)/f   Equation 8

In Equation 8, f is a focal length of the camera 3, and D_(t) is adistance between the camera 3 and the mobile target 2 that can bemeasured via, for example, a telemeter on-board the aircraft 1 anddirected toward the mobile target 2.

T_(C) ^(G) of Equation 7 is a transformation matrix (e.g., a rotationand translation matrix) and {circumflex over (t)}_(C) is translationvector for transitioning between a reference system {G} of globalcoordinates of a Global Reference System and a local reference system{C} of the camera 3 of FIG. 2. Further, C of Equation 7 is an intrinsicmatrix of the camera 3 that is defined below in Equation 9:

$\begin{matrix}{C = \begin{bmatrix}\alpha_{u} & \gamma & u_{0} \\0 & \alpha_{v} & v_{0} \\0 & 0 & 1\end{bmatrix}} & {{Equation}\mspace{14mu} 9}\end{matrix}$

In Equation 9 provided above, u₀ and v₀ represent a central point of theimage obtained by the camera 3, α_(u) and α_(v) represent scalingfactors of the axes û and {circumflex over (v)} of the image obtained bythe camera 3, and γ represents a parameter that defines an orthogonalityand/or skewness between the axes û and {circumflex over (v)} of theplane of the image obtained by the camera 3.

Based on Equations 3-4 and 7-9 provided above, a vector, {right arrowover (Ψ)}_(c), defines a direction in which the aircraft 1 is pointing(e.g., toward the mobile target 2) from its current location and/orcoordinates. The vector, {right arrow over (Ψ)}_(c), is expressed in thelocal reference system of the camera 3 and is provided below in Equation10:

$\begin{matrix}{\overset{\rightarrow}{\Psi_{c}} = {{\frac{1}{s}*C^{- 1}*\begin{bmatrix}u_{t} \\v_{t} \\1\end{bmatrix}} = {\frac{1}{s}*\begin{bmatrix}\frac{1}{\alpha_{u}} & \frac{- \gamma}{\alpha_{u}*\alpha_{v}} & {\frac{\gamma*v_{0}}{\alpha_{u}*\alpha_{v}} - \frac{u_{0}}{\alpha_{v}}} \\0 & \frac{1}{\alpha_{v}} & \frac{- v_{0}}{\alpha_{v}} \\0 & 0 & 1\end{bmatrix}*\begin{bmatrix}u_{t} \\v_{t} \\1\end{bmatrix}}}} & {{Equation}\mspace{14mu} 10}\end{matrix}$

Equation 10 may be rewritten as provided below in Equation 11:

$\begin{matrix}{\overset{\rightarrow}{\Psi_{c}} = {\frac{1}{s}*\begin{bmatrix}\frac{v_{t} - v_{0}}{\alpha_{v}} \\{\frac{u_{t}}{\alpha_{v}} - \frac{u_{0}}{\alpha_{v}} - {\gamma*\frac{v_{t} - v_{0}}{\alpha_{u}*\alpha_{v}}}} \\1\end{bmatrix}}} & {{Equation}\mspace{14mu} 11}\end{matrix}$

Further, the coordinates of the vector, {right arrow over (Ψ)}_(c), thatare based in the local reference system {C} of the camera 3 can betranslated into coordinates of a vector, {right arrow over (Ψ)}_(G),that are based in the global coordinates {G} of the Global ReferenceSystem. The vector, {right arrow over (Ψ)}_(G), is provided below inEquation 12:

$\begin{matrix}{\overset{\rightarrow}{\Psi_{G}} = {\begin{bmatrix}\psi_{x} \\\psi_{y} \\\psi_{z}\end{bmatrix} = {R_{U}^{C}*R_{G}^{U}*\overset{\rightarrow}{\Psi_{C}}}}} & {{Equation}\mspace{14mu} 12}\end{matrix}$

In Equation 12, R_(U) ^(C) is a rotation matrix between a referencesystem of the camera 3 and a reference system of the aircraft 1, andR_(G) ^(U) is a rotation matrix between the reference system of theaircraft 1 and the Global Reference System.

In combination with Equations 11-12 provided above, a parametricequation provided below in Equation 13 is utilized to determine thetarget position (e.g., the first position) of the aircraft 1 in theglobal coordinates {G} of the Global Reference System at which thecamera 3 is pointed toward the mobile target 2:

${\begin{bmatrix}x_{w} \\y_{w} \\z_{w}\end{bmatrix} = {\begin{bmatrix}x_{t} \\y_{t} \\z_{t}\end{bmatrix} + {\lambda*\overset{\rightarrow}{\Psi_{G}}}}},$

Thus, Equation 13 provided above enables the first position of theaircraft 1 that minimizes the area of uncertainty 5 to be determinedwhen

$\begin{bmatrix}x_{t} \\y_{t} \\z_{t}\end{bmatrix}\quad$

{right arrow over (Ψ)}_(G), and λ are known and/or determined.

To determine λ, a quadric or ruled space is defined to express the areaof uncertainty 5 associated with the location of the mobile target 2based on the state vector of Equation 1. The quadric or ruled space isprovided below in Equation 14:

x ^(T) *Q*x+p*x+r=0,

where:

$\begin{matrix}{{x = \begin{bmatrix}x \\y \\z\end{bmatrix}},{Q = \begin{bmatrix}q_{11} & q_{12} & q_{13} \\q_{21} & q_{22} & q_{23} \\q_{31} & q_{32} & q_{33}\end{bmatrix}},{p = \begin{bmatrix}p_{1} \\p_{2} \\p_{3\;}\end{bmatrix}},} & {{Equation}\mspace{14mu} 14}\end{matrix}$

and r is a constantIn Equation 14 provided above, the terms are inherent to an equationthat defines a quadric. For example, if the quadric is an ellipsoid, thecondition det(Q)>0 is set on the determinant of the matrix Q, which isan invertible matrix.

To determine a direction and/or module of a major axis 6 and/or minoraxis 7 of the ellipsoid of the area of uncertainty 5 corresponding tothe mobile target 2, Equation 14 may be rewritten with reference to acenter of the ellipsoid as provided below in Equation 15:

(x−k)^(T) *RDR ^(T)*(x−k)=1   Equation 15

In Equation 15 provided above, k represents the center of the ellipsoid(e.g., the coordinates of the mobile target 2), R represents a rotationmatrix, and D is a diagonal matrix. Based on Equations 15 providedabove, Equation 14 can be rewritten as provided below in Equation 16:

(x−k)^(T) *Q*(x−k)=x ^(T) *Q*x−2*k ^(T) *Q*x+k ^(T) *Q*k=(x ^(T) *Q*x+p^(T) *x+r)−(2*Q*k+p)^(T) *x+(k ^(T) *Q*k−r)=−(2*Q*k+p)^(T) *x+(k ^(T)*Q*k−r)   Equation 16

Further, the center of the ellipsoid, k, can be determined as providedbelow in Equation 17:

k=−Q ⁻¹ *p/2   Equation 17

Equation 17 can be rewritten as provided below in Equation 18:

k ^(T) *Q*k=p ^(T) *Q ⁻¹ *p/4   Equation 18

Further, Equation 18 can be rewritten as provided below in Equation 19:

(x−k)^(T) *Q*(x−k)=p ^(T) *Q ⁻¹ *p/4−r   Equation 19

By dividing Equation 19 by the scalar, r, Equation 19 can be rewrittenas provided below in Equation 20:

$\begin{matrix}{\Phi = \frac{Q}{\left( {{p^{T} \star Q^{- 1} \star {p/4}} - r} \right)}} & {{Equation}\mspace{14mu} 20}\end{matrix}$

In Equation 20, represents Φ a quadric. Further, Equation 20 can berewritten as provided below in Equation 21:

(x−k)^(T)*Φ*(x−k)=1   Equation 21

In Equation 21 provided above, a symmetric matrix of the quadric, Φ, isfactored utilizing an eigenvalue decomposition such that Φ=R*D*R^(T), inwhich R is the rotation matrix and D is the diagonal matrix. Forexample, a main diagonal of the diagonal matrix, D, is composed ofpositive factors.

In some examples, terrain on which the mobile target 2 is positionedand/or located is known via a Geographic Information System (GIS)database. In such examples, there is substantially no uncertainty of theposition of the mobile target in a z-axis. Further, in such examples,uncertainty in an x-y plane (e.g., a horizontal plane) is determined toidentify the area of uncertainty 5 of the mobile target 2. In examplesin which the uncertainty is an ellipsoid extending in the x-y plane, aneigenvalue decomposition is determined, calculated and/or otherwiseanalyzed to determine the major axis 6 and the minor axis 7 of theellipsoid of the area of uncertainty 5. For example, the quadric, Φ, acorresponding eigenvector, ω, and a corresponding eigenvalue, λ, can bewritten as provided below in Equation 22 when the eigenvalue, λ, is anon-null (e.g., non-zero) vector:

Φ*ω=λ*ω   Equation 22

In Equation 22 provided above, the eigenvalues are determined based onthe quadratic equation provided below in Equation 23:

det(Φ−λ*I)=0, where

Φ=(_(φ) ₂₁ ^(φ) ¹¹ _(φ) ₂₂ ^(φ) ¹² )   Equation 23

Based on Equation 23, the eigenvalues λ₁ and λ₂ are determined utilizingEquation 24 provided below:

$\begin{matrix}{\lambda_{1,2} = \frac{\left( {\phi_{11} + \phi_{22}} \right) \pm \sqrt[2]{\left( {\phi_{11} - \phi_{22}} \right)^{2} + {4 \star \phi_{12}^{2}}}}{2}} & {{Equation}\mspace{14mu} 24}\end{matrix}$

For example, the eigenvalues λ₁ and λ₂ that are determined utilizingEquation 24 are utilized in Equation 13 to determine the first positionthat minimizes the area of uncertainty 5.

Further, the eigenvalues λ₁ and λ₂ are utilized to determine the area ofuncertainty 5 associated with the mobile target 2. The eigenvalues λ₁and λ₂ can be expressed in Equation 25 and Equation 26 as providedbelow:

$\begin{matrix}{{\Phi \star \omega_{1}} = {\lambda_{1} \star \omega_{1}}} & {{Equation}\mspace{14mu} 25} \\{{\Phi \star \omega_{2}} = {\lambda_{2} \star \omega_{2}}} & {{Equation}\mspace{14mu} 26}\end{matrix}$

In Equation 25, ω₁ represents the major axis 6 of the ellipsoid of thearea of uncertainty. In Equation 26, ω₂ represents the major axis 6 ofthe ellipsoid of the area of uncertainty. Further, Equations 25 and 26are rewritten as provided below in Equation 27:

Φ*R=R*D   Equation 27

In Equation 27, R=[ω₁ ω₂] in which the columns are eigenvectors ofunitary length and D=diag(λ₁;λ₂). By multiplying the right side of theprevious equation by the matrix R^(T), Equation 28 as provided below isformed:

Φ=R*D*R ^(T)   Equation 28

Thus, the equation that defines the ellipsoid of the area of uncertainty5 is rewritten in Equation 29 as provided below:

(x−k)^(T)*Φ*(x−k)=1   Equation 29

In Equation 29, the quadric, Φ, corresponds to the covariance matrix Σas provided below in Equation 30:

Φ=¼*Σ⁻¹   Equation 30

The covariance matrix Σ of Equation 30 is periodically calculated and/ordetermined, for example, by a processor of the aircraft 1.

Based on the covariance matrix Σ, Equation 23, and Equation 30, themajor axis, ω₁, (the major axis 6 in FIGS. 3 and 4) and the minor axis,ω₂, (the minor axis 7 in FIGS. 3 and 4) of the area of uncertainty 5 aredetermined. For example, if φ₁₁≧φ₂₂, the major axis, ω₁, is determinedutilizing Equation 31 provided below and the minor axis, ω₂, isdetermined utilizing Equation 32 provided below:

$\begin{matrix}{\omega_{1} = {\begin{bmatrix}\omega_{1x} \\\omega_{1y}\end{bmatrix} = \begin{bmatrix}\frac{\lambda_{1} - \phi_{22}}{\sqrt{\left( {\lambda_{1} - \phi_{22}} \right)^{2} + \phi_{12}^{2}}} \\\frac{\phi_{12}}{\sqrt{\left( {\lambda_{1} - \phi_{22}} \right)^{2} + \phi_{12}^{2}}}\end{bmatrix}}} & {{Equation}\mspace{14mu} 31} \\{\omega_{2} = {\begin{bmatrix}\omega_{2x} \\\omega_{2y}\end{bmatrix} = \begin{bmatrix}\frac{\phi_{12}}{\sqrt{\left( {\lambda_{1} - \phi_{11}} \right)^{2} + \phi_{12}^{2}}} \\\frac{\lambda_{1} - \phi_{11}}{\sqrt{\left( {\lambda_{1} - \phi_{11}} \right)^{2} + \phi_{12}^{2}}}\end{bmatrix}}} & {{Equation}\mspace{14mu} 32}\end{matrix}$

Otherwise, if φ₁₁<φ₂₂, the major axis, ω₁, is determined utilizingEquation 33 provided below and the minor axis, ω₂, is determinedutilizing Equation 33 provided below:

$\begin{matrix}{\omega_{1} = {\begin{bmatrix}\omega_{1x} \\\omega_{1y}\end{bmatrix} = \begin{bmatrix}\frac{\phi_{12}}{\sqrt{\left( {\lambda_{1} - \phi_{11}} \right)^{2} + \phi_{12}^{2}}} \\\frac{\lambda_{1} - \phi_{11}}{\sqrt{\left( {\lambda_{1} - \phi_{11}} \right)^{2} + \phi_{12}^{2}}}\end{bmatrix}}} & {{Equation}\mspace{14mu} 33} \\{\omega_{2} = {\begin{bmatrix}\omega_{2x} \\\omega_{2y}\end{bmatrix} = \begin{bmatrix}\frac{\lambda_{1} - \phi_{22}}{\sqrt{\left( {\lambda_{1} - \phi_{22}} \right)^{2} + \phi_{12}^{2}}} \\\frac{\phi_{12}}{\sqrt{\left( {\lambda_{1} - \phi_{22}} \right)^{2} + \phi_{12}^{2}}}\end{bmatrix}}} & {{Equation}\mspace{14mu} 34}\end{matrix}$

The axes of the area of uncertainty 5 can change dynamically as thedetermined location of the mobile target 2 changes based on new and/oradditional data and/or information obtained regarding the mobile target2 and/or the aircraft 1.

Returning to determining the first position of the aircraft 1 thatminimizes the area of uncertainty 5, the first position can bedetermined utilizing Equations 11-13 and 24 given some constraintsand/or assumptions. For example, a constraint that the mobile target 2is to be centered in the image obtained by the camera 3 of the aircraftis represented by Equation 35 provided below:

$\begin{matrix}{{\overset{\rightarrow}{m}}_{t} = {\begin{bmatrix}u_{t} \\v_{t}\end{bmatrix} = \begin{bmatrix}{w/2} \\{h/2}\end{bmatrix}}} & {{Equation}\mspace{14mu} 35}\end{matrix}$

In Equation 35, w represents a number of pixels of a width of the imageobtained by the camera 3, and h represents a number of pixels of aheight of the image obtained by the camera 3. The values of Equation 35are utilized to determine the values of Ψ_(cx) and Ψ_(cy) in Equation11, which are utilized to determine x_(w) and y_(w) of the location thatminimizes the area of uncertainty 5.

Additionally or alternatively, another constraint requires that z_(w) ofthe location that minimizes the area of uncertainty 5 is assumed toequal z_(π). z_(π) represents an altitude of the aircraft 1 that isdetermined based on, for example, an instantaneous field of view (IFOV)of the camera 3 and a size of the mobile target 2. In some examples,z_(π) equals the size of the mobile target 2 divided by the IFOV of thecamera 3. For example, based on the constraint regarding the altitude ofthe aircraft 1, Equation 13, which represents the first position of theaircraft 1, can be rewritten as provided below in Equation 36:

$\begin{matrix}{\begin{bmatrix}x_{w} \\y_{w} \\z_{w}\end{bmatrix} = \begin{bmatrix}{x_{t} + {\left( z_{\pi} \right) \star \frac{\psi_{x}}{\psi_{z}}}} \\{y_{t} + {\left( z_{\pi} \right) \star \frac{\psi_{y}}{\psi_{z}}}} \\z_{\pi}\end{bmatrix}} & {{Equation}\mspace{14mu} 36}\end{matrix}$

Further, another constraint may require the camera 3 to be aligned witha plane of symmetry of the aircraft 1 and inclined at the angle θ (FIG.2) with respect to the fuselage 4 of the aircraft 1. As a result, therotational matrix, R_(U) ^(C), which is utilized to determine the firstlocation of the aircraft 1, is represented by the matrix provided belowin Equation 37:

$\begin{matrix}{R_{U}^{C} = \begin{bmatrix}0 & {\sin \mspace{11mu} \theta} & {\cos \mspace{11mu} \theta} \\1 & 0 & 0 \\0 & {\cos \mspace{11mu} \theta} & {{–sin}\mspace{11mu} \theta}\end{bmatrix}} & {{Equation}\mspace{14mu} 37}\end{matrix}$

To determine the rotational matrix, R_(G) ^(U), other constraints and/orassumptions are made. For example, the roll angle, γ_(w), and the pitchangle, β_(w), are assumed to equal 0. Further, the yaw angle, α_(w), ofthe camera 3 is focused on the mobile target 2 in a direction that isperpendicular to the major axis, ω₁, of the area of uncertainty 5. Theyaw angle, α_(w), is determined based on Equation 38 provided below:

$\begin{matrix}{\alpha_{w} = {\frac{\pi}{2} - {\tan^{- 1}\left( {- \frac{\omega_{1x}}{\omega_{1y}}} \right)}}} & {{Equation}\mspace{14mu} 38}\end{matrix}$

For example, in Equation 38, an angle of 0 degrees is directed to thenorth and a positive angle is clockwise from the north.

Based on the constraints regarding the roll angle, γ_(w), the pitchangle, β_(w), and the yaw angle, α_(w), the rotational matrix, R_(G)^(U), which is utilized to determine the first location of the aircraft1, is represented by the matrix provided below in Equation 37:

$\begin{matrix}{R_{G}^{U} = \begin{bmatrix}{\sin \mspace{11mu} \alpha_{w}} & {\cos \mspace{11mu} \alpha_{w}} & 0 \\{\cos \mspace{11mu} \alpha_{w}} & {{–sin}\mspace{11mu} \alpha_{w}} & 0 \\0 & 0 & {–1}\end{bmatrix}} & {{Equation}\mspace{14mu} 39}\end{matrix}$

For example, the yaw angle, α_(w), of Equation 39 is calculated based onEquation 38.

Based on the following constraints, two positions that minimize the areaof uncertainty 5 are determined because eigenvalues λ of Equation 24 isa quadratic equation. The aircraft 1 is to move to the one of the twopositions,

${\begin{bmatrix}x_{w\; 1} \\y_{w\; 1}\end{bmatrix}\mspace{14mu} {{or}\mspace{14mu}\begin{bmatrix}x_{w\; 2} \\y_{w\; 2}\end{bmatrix}}}\mspace{14mu}$

that is closest to the current coordinates,

$\quad{\begin{bmatrix}x \\y\end{bmatrix}.}$

The closer of the two positions of the aircraft 1 is determined based onEquation 40 provided below:

$\begin{matrix}{{{{{If}\mspace{20mu} \sqrt{\left( {x_{w\; 1} - x_{0}} \right)^{2} + \left( {y_{w\; 1} - y_{0}} \right)^{2}}} \leq \sqrt{\left( {x_{w\; 2} - x_{0}} \right)^{2} + \left( {y_{w\; 2} - y_{0}} \right)^{2}}},\mspace{79mu} {{{{then}\mspace{14mu} y_{w\; 1}} - y_{t}} = {{- \frac{\omega_{1x}}{\omega_{1y}}}\left( {x_{w\; 1} - x_{t}} \right)}},\mspace{79mu} {{{{otherwise}\mspace{14mu} y_{w\; 2}} - y_{t}} = {{- \frac{\omega_{1x}}{\omega_{1y}}}\left( {x_{w\; 2} - x_{t}} \right)}}}\mspace{14mu}} & {{Equation}\mspace{14mu} 40}\end{matrix}$

Thus, if the first equation of Equation 40 is true, the aircraft 1 is tomove to

$\quad\begin{bmatrix}x_{w\; 1} \\y_{w\; 1}\end{bmatrix}$

to deduce the area of uncertainty 5. Otherwise, the aircraft 2 is tomove to

$\quad\begin{bmatrix}x_{w\; 2} \\y_{w\; 2}\end{bmatrix}$

to deduce the area of uncertainty 5.

FIG. 4 illustrates the aircraft 1 and another aircraft 8 positioned toreduce the area of uncertainty 5 of the mobile target 2 in accordancewith the teachings of his disclosure. As illustrated in FIG. 4, theaircraft 8 includes a camera 9. The aircraft 8 and the camera 9 of FIG.4 are substantially similar or identical to the aircraft 1 and thecamera 3, respectively, of FIGS. 3-4. Because the aircraft 1 and thecamera 3 are described in detail in connection with FIG. 3, somecharacteristics of the aircraft 1 and the camera 3 are not described infurther detail below.

In the illustrated example, the camera 3 of the aircraft 1 is focused onthe mobile target 2 in a direction that is perpendicular to the majoraxis 6 (the major axis, ω₁, in the above equations) of the area ofuncertainty 5, and the camera 9 of the aircraft 8 is focused on themobile target 2 in a direction that is perpendicular to the minor axis 7(the minor axis, ω₂, in the above equations) of the area of uncertainty5. For example, the aircraft 1 is in the first position that isorthogonal to the major axis 6 along the minor axis 7 and the aircraft 8is in a second position (e.g., a second principal position) along themajor axis 6 orthogonal to the minor axis 7. The aircraft 1, 8 track themobile target 2 from substantially perpendicular positions and/ordirections to further reduce the area of uncertainty 5 related to thedetermined position of the mobile target 2.

FIG. 5 depicts the Global Reference System, {G}, utilized to determinecoordinates of the aircraft 1 (FIGS. 1-4), the aircraft 8 (FIG. 4)and/or the mobile target 2 (FIGS. 3-4). In the illustrated example, the{circumflex over (x)} axis points to the east, the ŷ axis points to thenorth, and the {circumflex over (z)} axis is extends between the centerof the Earth and the mobile target 2.

FIG. 6 is a flowchart representative of an example method 60 to positionaircraft(s) to reduce an area of uncertainty of coordinates of a mobiletarget in accordance with the teachings herein. Although the examplemethod 60 is described with reference to the flowchart illustrated inFIG. 6, many other methods for positioning the aircraft(s) to reduce thearea of uncertainty of the coordinates of the mobile target mayalternatively be used. For example, the order of execution of the blocksmay be changed, and/or some of the blocks described changed, eliminated,and/or combined.

Further, the blocks of the example method 60 may be implemented byexecuting corresponding instructions (e.g., first instructions, secondinstructions, third instructions, etc.) via a processor. For example,the processor is hardware of the example aircraft 1 and/or the exampleaircraft 8. The processor can be implemented by one or more integratedcircuits, logic circuits, microprocessors or controllers from anydesired family or manufacturer. In some examples, the processor includesa local memory (e.g., a cache). In some examples, the memory includesvolatile memory and/or non-volatile memory. Example volatile memory maybe implemented by Synchronous Dynamic Random Access Memory (SDRAM),Dynamic Random Access Memory (DRAM), RAMBUS Dynamic Random Access Memory(RDRAM) and/or any other type of random access memory device. Examplenon-volatile memory may be implemented by flash memory and/or any otherdesired type of memory device. Access to the memory is controlled, forexample, by a memory controller. In some examples, the memory is atangible computer readable storage medium such as a flash memory, aread-only memory (ROM), a cache, a random-access memory (RAM) and/or anyother storage device or storage disk in which information is stored forany duration (e.g., for extended time periods, permanently, for briefinstances, for temporarily buffering, and/or for caching of theinformation). As used herein, the term tangible computer readablestorage medium is expressly defined to include any type of computerreadable storage device and/or storage disk and to exclude propagatingsignals and transmission media. As used herein, “tangible computerreadable storage medium” and “tangible machine readable storage medium”are used interchangeably. In some examples, the memory is anon-transitory computer and/or machine readable medium such as a flashmemory, a read-only memory, a cache, a random-access memory and/or anyother storage device or storage disk in which information is stored forany duration (e.g., for extended time periods, permanently, for briefinstances, for temporarily buffering, and/or for caching of theinformation). As used herein, the term non-transitory computer readablemedium is expressly defined to include any type of computer readablestorage device and/or storage disk and to exclude propagating signalsand transmission media.

The method 60 for positioning aircraft(s) to reduce an area ofuncertainty of coordinates of a mobile target is discussed in connectionwith the aircraft 1 of FIGS. 1-4, the aircraft 8 of FIG. 4, the mobiletarget 2 of FIGS. 3-4 and/or the area of uncertainty 5 of FIGS. 3-4.Further, because the method 60 may refer to the aircraft 1 of FIGS. 1-4,the aircraft 8 of FIG. 4, the mobile target 2 of FIGS. 3-4 and/or thearea of uncertainty 5 of FIGS. 3-4, components identified in FIGS. 1-4having functions substantially similar or identical to the functions ofcomponents described below will not be described in detail again.Instead, the same reference numbers will be used for like structures.

The example method 60 disclosed herein starts at block 61 by a camera(e.g., the camera 3 of FIGS. 2-4, the camera 9 of FIG. 4) mounted to anaircraft (e.g., the aircraft 1 of FIGS. 1-4, the aircraft 8 of FIG. 4)obtaining an image of a mobile target (e.g., the mobile target 2 ofFIGS. 3-4). At block 62, the method 60 includes determining whetherthere is another image to be obtained. Blocks 61, 62 are repeated untilall of the images to be obtained of the mobile target are obtained.

At block 63, current coordinates of a first aircraft (e.g., the aircraft1 of FIGS. 1-4, the aircraft 8 of FIG. 4) and characteristics of a firstcamera (e.g., the camera 3 of FIGS. 2-4, the camera 9 of FIG. 4) of thefirst aircraft are obtained. At block 64, the method 60 includesdetermining whether there is another aircraft. If there is anotheraircraft, current coordinates of a second aircraft (e.g., the aircraft 1of FIGS. 1-4, the aircraft 8 of FIG. 4) and characteristics of a secondcamera (e.g., the camera 3 of FIGS. 2-4, the camera 9 of FIG. 4) of thesecond aircraft are obtained (block 65).

Upon determining that there is not another aircraft or upon obtainingthe coordinates of the second aircraft and the characteristics of thecorresponding camera, coordinates of the mobile target are determinedbased on the image(s) obtained from the camera, the coordinates of theaircraft(s), and the characteristics of the camera(s) (block 66).Further, an area of uncertainty (e.g., the area of uncertainty 5 ofFIGS. 3-4) is determined for the coordinates of the mobile target (block67).

At block 68, the first aircraft is positioned to be orthogonal to amajor axis (e.g., the major axis 6 of FIGS. 3-4) of the area ofuncertainty to reduce the area of uncertainty of the mobile target. Forexample, the first aircraft is positioned to be along a minor axis(e.g., the minor axis 7 of FIGS. 3-4) of the area of uncertainty. Atblock 69, the method 60 determines if there is another aircraft toposition. If there is not another aircraft, the method 60 ends. If thereis another aircraft to be positioned, the second aircraft is positionedto be orthogonal to the minor axis of the area of uncertainty to furtherreduce the area of uncertainty of the mobile target (block 70). Forexample, the second aircraft is positioned to be along the major axis ofthe area of uncertainty. In some examples, the area of uncertainty, thefirst position of the first aircraft and/or the second position of thesecond aircraft is changed and/or updated dynamically as new data and/orinformation is obtained regarding the aircraft(s) and/or the mobiletarget.

Although certain example apparatus and methods have been describedherein, the scope of coverage of this patent is not limited thereto. Onthe contrary, this patent covers all methods, apparatus and articles ofmanufacture fairly falling within the scope of the amended claims eitherliterally or under doctrine of equivalents.

What is claimed is:
 1. A method comprising: identifying, by executingfirst instructions via a processor, a mobile target in an image obtainedby a camera mounted on a first aircraft; obtaining, by executing secondinstructions via the processor, current coordinates of the firstaircraft; determining, by executing third instructions via theprocessor, coordinates of the mobile target based on the coordinates ofthe first aircraft and the image, the coordinates of the mobile targetbeing within an area of uncertainty; and determining, by executingfourth instructions via the processor, a first position for the firstaircraft that reduces the area of uncertainty of the coordinates of themobile target.
 2. The method of claim 1, wherein obtaining the currentcoordinates of the first aircraft includes receiving data via a globalpositioning sensor of the first aircraft.
 3. The method of claim 1,wherein determining the first position to reduce the area of uncertaintyincludes determining the first position to minimize the area ofuncertainty.
 4. The method of claim 1, wherein determining the firstposition for the first aircraft that reduces the area of uncertainty isbased on at least one of the coordinates of the mobile target, thecurrent coordinates of the first aircraft, the image obtained by thecamera, a focal length of the camera, an angle of the camera relative toa fuselage of the first aircraft, or a yaw angle of the first aircraft.5. The method of claim 1, wherein determining the coordinates of themobile target based on the coordinates of the first aircraft and theimage includes determining 3-dimensional coordinates of the mobiletarget, the coordinates of the first aircraft being 3-dimensional, theimage obtained by the camera being 2-dimensional.
 6. The method of claim1, further including determining a distance between the first aircraftand the mobile target to determine the coordinates of the mobile targetbased on the coordinates of the first aircraft and the image.
 7. Themethod of claim 1, further including determining the area of uncertaintyof the coordinates of the mobile target, the area of uncertainty havingan elliptical shape and including a major axis and a minor axis.
 8. Themethod of claim 7, wherein determining the first position of the firstaircraft that reduces the area of uncertainty includes determining aposition orthogonal to the major axis and along the minor axis.
 9. Themethod of claim 8, wherein determining the first position of the firstaircraft includes identifying two positions along the minor axis thatminimize the area of uncertainty and determining which one of the twopositions is closest to the current coordinates of the first aircraft.10. The method of claim 7, wherein identifying the mobile target isfurther based on a second image obtained by a second camera mounted on asecond aircraft.
 11. The method of claim 10, further including:obtaining current coordinates of the second aircraft; determiningcoordinates of the mobile target further based on the coordinates of thesecond aircraft and the second image; and determining a second positionfor the second aircraft that reduces the area of uncertainty of thecoordinates of the mobile target.
 12. The method of claim 11, whereindetermining the second position of the second aircraft that reduces thearea of uncertainty includes determining a position orthogonal to theminor axis and along the major axis of the area of uncertainty.
 13. Themethod of claim 1, further including moving the first aircraft to thefirst position to enable the camera of the first aircraft to obtain asecond image, the mobile target being centered in the second image. 14.The method of claim 1, further including determining at least one of asize, a color, or a velocity of the mobile target to determine aclassification of the mobile target.
 15. An apparatus comprising: acamera mounted to an aircraft to obtain an image; a processor of theaircraft to: identify a mobile target in the image obtained by thecamera; obtain current coordinates of the aircraft; determinecoordinates of the mobile target based on the coordinates of theaircraft and the image, the coordinates of the mobile target beingwithin an area of uncertainty; determine a position for the aircraftthat reduces the area of uncertainty of the coordinates of the mobiletarget; and instruct the aircraft to move to the position to reduce thearea of uncertainty.
 16. The apparatus of claim 15, further including aglobal positioning sensor of the aircraft, the processor to obtain thecurrent coordinates of the aircraft via the global positioning sensor.17. The apparatus of claim 15, wherein the camera has a focal length andis positioned at an angle relative a fuselage of the aircraft, theprocessor of the aircraft determines the position of the aircraft thatreduces the area of uncertainty based on at least one of the focallength or the angle of the camera.
 18. The apparatus of claim 15,wherein the processor determines the area of uncertainty of thecoordinates of the mobile target, the area of uncertainty having anelliptical shape including a major axis and a minor axis, the positionof the aircraft that reduces the area of uncertainty is orthogonal tothe major axis and along the minor axis.
 19. The apparatus of claim 15,wherein the camera is to obtain a second image when the aircraft is atthe position that reduces the area of uncertainty, the mobile targetbeing centered in the second image.
 20. An apparatus comprising: meansfor obtaining an image mounted to an aircraft; means for determining aposition of the aircraft to identify a mobile target in an imageobtained by the means for obtaining an image, obtain current coordinatesof the aircraft, determine coordinates of the mobile target and an areaof uncertainty based on the coordinates of the aircraft and the image,and determine a position for the aircraft that reduces the area ofuncertainty; and means for moving the aircraft to the position to reducethe area of uncertainty.